Idea

In order to express this kinship of these different cohomological theories, I formulated the notion of “motive” associated to an algebraic variety. By this term I want to suggest that it is the “common motive” (or “common reason”) behind this multitude of cohomological invariants attached to an algebraic variety, or indeed, behind all cohomological invariants that are a priori possible. (Grothendieck, Recoltes et Semailles)

The similarity of the behaviour of various cohomologies of varieties over a field suggests that there is a universal one among them with values in an intermediate abelian category, called the category of motives. The idea is that to every variety XX is associated a motive M(X)M(X), such that every good cohomology theory factors through the functor MM. (Here not every motive is supposed to be the image of a single variety.)

One distinguishes a theory of pure motives for smooth projective varieties from a more general theory of mixed motives for arbitrary smooth varieties. So far, pure motives and mixed motives have only been defined conditionally. However there are several equivalent definitions of a triangulatedtensor category which has all conjectured structural properties of the derived category of mixed motives (except for the t-structure which would make it a derived category).

LL-functions (and ζ\zeta-functions in particular) of varieties are also invariants of their motives. The Langlands program indirectly involves motives; in particular its essential part can be expressed as a general modularity conjecture relating LL-functions to automorphic functions. Most of the deep properties of elliptic curves are of motivic nature, and in particular a major step of the proof of Fermat's last theorem by Wiles and Taylor can be interpreted as a proof of a special case of the modularity conjecture (for elliptic curves).

Constructions of the abelian category of mixed motives

There is no generally accepted construction of a ℚ\mathbb{Q}-linear abelian category of mixed motives, and its existence remains conjectural. However, there exist candidate and conditional constructions which are useful in practice.

Note that “the” abelian category of mixed motives depends on choosing a base schemeSS, and one speaks of motives (or motivic sheaves) over SS. Traditionally, SS is the spectrum of a field, often of characteristic zero.

Deligne motives

Pierre Deligne gave a definition of a category of mixed motives over number fields as compatible systems of realizations, essentially bundling together all the structure that mixed motives should give rise to. This approach automatically yields a ℚ\mathbb{Q}-Tannakian category of mixed motives with all the desired realization functors (Betti, ll-adic, de Rham, and crystalline). See Deligne for details.

As the heart of a t-structure on the derived category of mixed motives

Deligne first suggested that it might be easier to define the derived categoryDM(S,ℚ)DM(S,\mathbb{Q}) of the hypothetical abelian category of mixed motives. Once this is done, one can in principle recover the abelian category as the heart of a t-structure on DM(S,ℚ)DM(S,\mathbb{Q}). It is now well-understood what the triangulated categoryDM(S,ℚ)DM(S,\mathbb{Q}) is over any base scheme (see below). The hypothetical t-structure on DM(S,ℚ)DM(S,\mathbb{Q}) whose heart is the abelian category of mixed motives over SS is called the motivic t-structure.

While the derived category of mixed motives can also be defined with integral rather than rational coefficients, Voevodksy observed that the derived category of integral motives cannot have a motivic t-structure (Voevodsky, Prop. 4.3.8). Thus, the abelian category of motives always refers to motives with rational coefficients.

Constructions of the derived category of mixed motives

The derived category of the hypothetical abelian category of mixed motives has been unconditionally defined over any Noetherian scheme. The first definition was proposed by Voevodsky in the mid 1990s. Since then, several other definitions were formulated: one by Morel, one by Ayoub, and one by Cisinski and Déglise. The latter three are equivalent and support a full-fledged formalism of six operations. However, they are only known to be equivalent to Voevodsky’s definition over excellent? and geometrically unibranch? schemes.

On the other hand, Voevodsky’s definition is the only one among these four which also makes sense with integral coefficients rather than rational coefficients. Recently, Spitzweck proposed a definition of the category of integral motives over general base schemes which also supports a formalism of six operations. It is known to agree with Voevodsky’s definition for fields of characteristic zero. Rationally, however, it agrees with the Morel/Ayoub/Cisinski-Déglise definition over any base scheme.

morphisms SmCorS(X,Y)SmCor_S(X,Y) are the abelian group of cycles on the fiber productX×SYX \times_S Y that are “universally integral relative to XX” and each of whose components are finite and and surjective over XX.

Let ϵ:𝔾m∧𝔾m→𝔾m∧𝔾m\epsilon : \mathbb{G}_m\wedge \mathbb{G}_m\to \mathbb{G}_m\wedge \mathbb{G}_m be the transposition. In the stable motivic homotopy categorySH(S)SH(S) this becomes an endomorphism of the motivic sphere spectrum S0S^0 such that ϵ2=1\epsilon^2=1. Rationally (or even away from 2), we obtain a pair of idempotent elements

Definition

SH(S)ℚ+SH(S)_{\mathbb{Q}_+} is the stable (∞,1)(\infty,1)-category of Morel motives.

In other words, a Morel motive is a rational stable motivic homotopy type on which ϵ\epsilon acts as −1-1.

The Hopf elementη∈π1,1(S0)\eta\in \pi_{1,1}(S^0) is the stabilization of the algebraic Hopf fibration𝔸2−0→ℙ1\mathbb{A}^2-0\to\mathbb{P}^1 over SS. Morel motives can also be characterized as those rational stable motivic homotopy types that are acted on trivially by the Hopf element.

We have ϵ=−1\epsilon=-1 if and only if −1-1 is a sum of squares in all the residue fields of SS, in which case SH(S)ℚ=SH(S)ℚ+SH(S)_{\mathbb{Q}}= SH(S)_{\mathbb{Q}_+}. Thus, the other summand SH(S)ℚ−SH(S)_{\mathbb{Q}_-} only appears over formally real fields. It is called the category of Witt motives.

As homotopy invariant étale sheaves without transfers (Ayoub motives)

According to Ayoub, the stable (∞,1)(\infty,1)-category of motives over a scheme SS can be constructed in the same way as the stable motivic homotopy category SH(S)SH(S), with two variations:

Definition

The stable (∞,1)(\infty,1)-category of Beilinson motives is the (∞,1)(\infty,1)-category of modules over HBH_B. Equivalently, it is the full subcategory of SH(S)ℚSH(S)_{\mathbb{Q}} consisting of HBH_B-local objects.

Cisinski and Déglise have shown that HBH_B is exactly the ++-summand Sℚ+0S^0_{\mathbb{Q}_+} of the rational motivic sphere spectrum, and hence that a Beilinson motive is the same thing as a Morel motive. They have also shown that Beilison/Morel motives are equivalent to Ayoub motives. Finally, they have shown that Beilinson motives are equivalent to rational Voevodsky motives DM(S,ℚ)DM(S,\mathbb{Q}) when SS is excellent? and geometrically unibranch?. Over such schemes, all four definitions of the derived category of mixed motives are therefore equivalent.

As modules over a spectrum representing motivic cohomology over SpecℤSpec \mathbb{Z} (Spitzweck motives)

One idea to define a category of integral motives with a formalism of six operations is to first define an E∞E_\infty motivic ring spectrum Mℤ∈SH(Specℤ)M_{\mathbb{Z}}\in SH(Spec \mathbb{Z}). If f:S→Specℤf: S\to Spec \mathbb{Z} is any scheme, we obtain an E∞E_\infty-algebra MS=f*(Mℤ)M_S = f^\ast(M_{\mathbb{Z}}) in SH(S)SH(S). The categories of modules over MSM_S for varying SS then inherit a complete formalism of six operations from SHSH.

Spitweck defined such an E∞E_\infty-algebra MℤM_{\mathbb{Z}} such that

if SS is smooth over a field, MSM_S is equivalent to Voevodsky’s motivic Eilenberg–Mac Lane spectrum HℤH\mathbb{Z},

Mℤ⊗ℚM_{\mathbb{Z}}\otimes\mathbb{Q} is the Beilinson motive HBH_B.

The stable (∞,1)(\infty,1)-category of MSM_S-modules is thus a well-behaved candidate for a derived category of integral motives, but it is only known to agree with Voevodsky’s definition when SS is a field of characteristic zero (by Rondigs-Ostvaer, Theorem 5.5).

Variations and extensions

Correspondences are interesting in noncommutative geometry of the operator algebra flavour. For example, KK-groups are in fact themselves sort of correspondences; Connes and Skandalis had an early reference very much paralleling some ideas from the algebraic world. More recently, motives in the operator algebraic setup have been approached by Connes, Marcolli and others.

In birational geometry, Bruno Kahn defined the appropriate version. In rigid analytic geometry, A1A^1-homotopy theory is replaced by B1B^1-homotopy theory and the appropriate analogue of the Voevodsky’s category of mixed motives has been constructed; the construction follows the same basic pattern.